### bricks in the wall

much of the point here is to “explain why”
the figure in the upper right… “pascal’s triangle”
works the way it does: students gladly learn
in minutes how to write out rows by adding
pairs taken from previous rows (according to
the formula in the lower left, though they
know it not; eventually they’ll have to be
encouraged to care but of course one need
not speak of thorny notational issues at
every single opportunity; when the student
is ready the code’ll have been there waiting
all along). the thing is: what the heck does
with “counting subsets”, then, eh? because
when we “use” these numbers in our little
probability-and-counting problems back
in class, we’ll be referring to “N choose R”
all the time: the number of ways to
“choose” a set of R things from a set of N
.
well, put like that…
the thing is more or less obviously
then to look carefully at arrangements-of-subsets
and see if we can piece out
“what comes from where”.

my best shot so far.
better with the handwaving of course.

1. Due to its simple construction by factorials, a very basic representation of Pascal’s triangle in terms of the matrix exponential can be given: Pascal’s triangle is the exponential of the matrix which has the sequence 1, 2, 3, 4, … on its subdiagonal and zero everywhere else.

good ol’ wikipedia.

2. okay suppose you’ve got to know about N
but all you know about is N-1. somebody
gives you N things anyhow and sez
“how many ways can i get R things
out of these here?”.

take one of the N things and paint it blue.

now there’s the N-1 colorless things and the blue one.
we’re supposed to pick a set of R things somehow.
okay… i say now there are *two* ways i might
go about getting such a set:
pick the *whole set of R things*
from the nonblue N-1…
there are {N-1 \choose R} ways to do this…
or
pick *only R-1 nonblue things* from
this same N-1, plus the blue thing for
the full quota of R things altogether:
there are {N-1 \choose R-1} ways
to do this.

so we’ll’ve seen that
${N \choose R} = {N-1 \choose R} + {N-1 \choose R-1}$.
(not the same formula as in
my handwritten display but
the same fact.)

i forget who put it to me like this
but this is the one that stuck.

3. vlorbik

back on the message of the day.
i did a new issue of the
“lectures without words”
series today: MathEdZine 0.8
(the K_n — K_4 remix).
it’s a classic “minicomic” format:
copy on both sides of a sheet
of typing paper, fold the long way
first, then fold again the short way,
staple along the spine (with an
ordinary desk stapler this will
entail folding over a few pages;
with a “longarm” stapler one
can just shove it it and bang down),
and trim it at the top. voila:
an eight-pager just like cynicalman.
without the brilliant art and
longpracticed production values.
anyhow, it’s the first zine i’ve made
with this particular “imposition” and
so i get a tiny bit of a thrill from it.
there’s also a sort of, oh-heck-give-up
to the whole thing though since i’d
never even thought of *doing* a remix
if the “unstaplebook” format of
the original 0.6 and 0.7 wasn’t
distracting almost every subject
i’ve had a chance to work with…

i’m glad i *did* the unstaplebooks.
part of the point *is* the very
what-the-heck-is-*this*
effect that the very format
of the document inspires…
among other things, this gives the “teacher”
(that would be me in the interviews i’ve had
but is always somebody else in my imagination)
a chance to bail out on the math
(i’ve-never-been-good-in-math
is just about as reliable a response
to being confronted with a MEdZ
as what-the-heck-is-this is) and
to the point for “educators” i suppose,
*self*-publishing and the DIY vibe
generally. teach a kid to write up
cool math facts and they’ll fish
for the rest of their life kinda thing.

but i’m laying off of ’em
(“unstaplebook” micro- and nano-
-zines) once i’ve filled in the
$\Bbb R$ and $\Bbb C$
issues at 0.4 and 0.5
i think.

4. vlorbik

the display in the photo
appears in both K_n
and the “remix” issue.

i meant to’ve said.
(why doesn’t somebody
just *interrupt* me
when i get going like that.)

5. I’m going to try to respond to kate(t)’s challenge by using binomial expansion to create pascal’s triangle. Maybe.

1
x | y
first put xs on the left: xx xy, then put ys on the left: yx yy

xx | xy yx | yy

first put xs on the left: xxx xxy xyx xyy, then put ys on the left: yxx yxy yyx yyy

xxx | xxy xyx yxx | xyy yxy yyx | yyy

counting, we get 1 2 1 0 + 0 1 2 1

like, no surprise, multiplication by 11.

Need to play and clean, and develop cooler graphics than I’ve ever done.

Jonathan

6. it looks to me like “pascal’s triangle”
… why do the binomial coefficients
behave the way they do…
is the heart of the matter for
kate’s problem. everything else
is easier for *me* to see, anyhow.

i’ve *also* got notes on how to
lay out answers and work ’em
by rote. i drilled several classes
in expanding (P + Q)^N
for probability-and-stix
applications for quite a bit
more than the standard presentations
would ever dream of doing just
so i’d feel like they’d learned
to do *something* calling for
honest algebraic manipulation
of binomial coefficients.
i had to cut out some of the
“pretend to learn a bunch of
technical terms for regurgitation”
stuff to do it though. oh the horror.

i’m unlikely to submit anything myself.
i’ve finally given in to the need to
develop cooler graphics than i’ve ever done.
with ink and paper and whatnot though…
making real objects in the real world.
computers are just way too expensive
and frustrating without massive
institutional support.

i can’t work this damn copier and
it breaks my heart.

• ## (Partial) Contents Page

Vlorbik On Math Ed ('07—'09)
(a good place to start!)